It was a bad afternoon for highly regarded young Everton defender John Stones, whose poor back-pass led to the 17th minute penalty from which Gylfi Sigurdsson gave Swansea the lead.
The home side soon equalised when Jack Cork touched Gareth Barry's flick into his own net, but a deflection off Stones from Andre Ayew's shot restored the Welsh club's lead before half-time.
They held out amid strong pressure in the second half, when Everton's leading scorer Romelu Lukaku twice went close and substitute Seamus Coleman twice missed badly, including from two yards out with the last kick of the match.
Achieving a second successive win for the first time this season moved Guidolin's team above Norwich City and Bournemouth into 15th place, increasingly confident of avoiding relegation back to the Championship after five years.
"This is our target," Guidolin told Sky Sports. "These (two) results are very important for us. We go day by day and step by step."
"What was important was the team spirit we showed," Ayew said. "The second half we knew was going to be difficult but sometimes when you show a lot of desire the luck comes to you, like with the last ball they could have scored from."
The defeat left Everton 12th, with their former Swansea manager Roberto Martinez rueing a slow start and missed chances.
"The scoreline was very hard to take," he said.
"We didn't start as well as we can, and were a little pedestrian. But with the second half reaction and the chances we had the scoreline could have been very, very different."
(Reporting by Steve Tongue, editing by Alan Baldwin)
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